Suppose we are holding a replicating portfolio $\Pi_t$ of long an option $f(S,t)$ and short some stock, so $$\Pi_t=f(S_t,t)-\Delta_t S_t$$ Suppose the stock follows geometric Brownian motion and pays continuous dividends at rate $q$, so $$d S_t = S_t((\mu-q)dt + \sigma dW_t$$ Naively, $$d\Pi_t = d f(S_t,t)-\Delta_t dS_t$$ However, because the stock pays a dividend, common sense and the literature tell us that $$d\Pi_t=df(S_t,t)-\Delta_t dS_t-\Delta_t S_t dt$$

Question: How do we rigorously arrive at the total derivative for $d\Pi_t$ which includes extra term $-\Delta_t S_t dt$, given that we know $\Pi_t=f(S_t,t)-\Delta_t S_t$, without appeal to common sense i.e., from the equations, without recalling the mechanics of how the stock works? Because, naively, I would not include the extra term, if I just knew the equation defining $\Pi_t$ and nothing about the mechanics of the stock.

  • $\begingroup$ Hi @Lars Ericson. What you are asking is little paradoxical to me. How can one write the equations of something "without knowing how it works"? More specifically, holding a stock is not a self-financing strategy: as a dividend gets paid this gets credited to your cash account. To avoid arbitrage opportunities, when that happens the price of stock gets amputated from the dividend amount (this is the $-qS_tdt$ part of the stock dynamics). The value of a dividend-paying stock thus differs from the value of a strategy consisting of holding that stock (in the latter dividends pop up as cash). $\endgroup$ – Quantuple Aug 16 '17 at 7:39
  • $\begingroup$ In other words: if you hold $\Delta_t$ stocks at $t$ (value $\Delta_t S_t$, which you hold for a small period of time $dt$ over which the stock pays a div yield $qdt$, the resulting position at $t+dt$ is worth $\Delta_t S_{t+dt} + \Delta _t S_t q dt$. Hence the infinitesimal variation of total value is indeed: $\Delta_t dS_t + \Delta_t S_t q dt = \Delta_t S_t (\mu dt + \sigma dW_t)$. Again the difference lies in the value of the stock vs. the value of a portfolio consisting of holding it. A difference which only exists if holding the asset is not a self-financing strategy. $\endgroup$ – Quantuple Aug 16 '17 at 7:46
  • $\begingroup$ This writeup has $dS=S((\mu - q) dt + \sigma dW)$ and $\Pi=f-\Delta S$ and then jumps to $d \Pi = df - \Delta dS - q S \Delta dt$ instead of $d \Pi=df-\Delta dS$. Since the total derivative $dS$ is already stated, I was confused how the extra term $q S \Delta dt$ comes into being, as the total derivative is already stated as $dS=S((\mu - q) dt + \sigma dW)$ , not $dS=S((\mu - q) dt + \sigma dW) + q S dt$. Basically the extra $q S dt$ popping up in the $d \Pi$ looked like magic, aside from "knowing how it works". $\endgroup$ – Lars Ericson Aug 17 '17 at 12:21
  • $\begingroup$ It's not a total derivative it's an Itô differential. It describes how the price of the stock will change over an infinitesimal period of time at order 1 in $dt$. And indeed you start with $\Pi = f - \Delta S$ and as time passes your make/lose $d\Pi = df - \Delta dS - \Delta qS dt$. The first term on the RHS is because the option price changed, the second term on the RHS is because the stock price has changed (according to the given $dS$)... the third is because this stock has paid dividends. $\Pi=\sum_i a_i X_i \to d\Pi = \sum_i a_i dX_i$ only holds for self-financing portfolios. $\endgroup$ – Quantuple Aug 17 '17 at 14:17
  • $\begingroup$ From a pure math perspective, ignoring the financial context, can you explain the difference between a total derivative and an Itô differential? I'm trying to separate the math from the application. $\endgroup$ – Lars Ericson Aug 17 '17 at 16:59

Let $S^0_t = e^{rt}$ be the money market account.

Consider you short a derivative $P_t$ and hedge it with cash and stock. $$ \Pi_t = \Delta^0_t S^0_t + \Delta_t S_t - P_t $$ At time $t+dt$, the portfolio is \begin{eqnarray*} \Pi_{t+dt} &=& \Delta^0_t S^0_{t+dt} + \Delta_t S_{t+dt} + \Delta_t q_t S_t dt - P_{t+dt} \end{eqnarray*} Where $\Delta_t q_t S_t dt$ corresponds to the dividend received. Then \begin{eqnarray*} \Pi_{t+dt} &=& (1+rdt)\Delta^0_t S^0_{t} + \Delta_t S_{t+dt} - P_{t+dt} \\ &=& (1+rdt)( \Pi_t - \Delta_t S_t + P_t ) + \Delta_t S_{t+dt} + \Delta_t q_t S_t - P_{t+dt} \\ &=& (1+rdt)( \Pi_t - \Delta_t S_t + P_t ) + \Delta_t S_{t+dt} + \Delta_t q_t S_t - P_{t+dt} \end{eqnarray*} This can be rewritten \begin{eqnarray*} d\Pi_{t}-r\Pi_t &=& \Delta_t (dS_{t} - (r-q)S_tdt) - (dP_{t} - rP_tdt) \end{eqnarray*}

Assume the stock price has dynamic $$ \frac{dS_t}{S_t} = \mu dt + \sigma dW^\mathbb{P}_t $$ under the real-world measure $\mathbb{P}$ (the drift could even be stochastic here). We search for the price in the form $P_t = P(t,S_t)$. Applying Ito, one finds \begin{eqnarray*} dP_{t} &=& \partial_tPdt + \partial_SP dS_t + \frac{1}{2}\sigma^2 S_t^2\partial^2_SP dt \end{eqnarray*} so \begin{eqnarray*} d\Pi_{t}-r\Pi_t &=& (\Delta_t - \partial_SP) dS_{t} - \Delta_t S_t (r-q)dt) - \partial_tPdt - \frac{1}{2}\sigma^2 S_t^2\partial^2_SP dt + rP(t,S_t)dt) \end{eqnarray*} In order to kill the stochastic term we should choose $\Delta_t = \partial_SP$. We end up with a risk-less portfolio with PnL \begin{eqnarray*} d\Pi_{t}-r\Pi_t &=& -\left(\partial_tP + (r-q)S_t \partial_SP + \frac{1}{2}\sigma^2 S_t^2\partial^2_SP - rP \right)dt \end{eqnarray*} By absence of arbitrage, this has to be 0 otherwise we could make a guaranteed profit without taking any risk. So the right hand side is the diffusion equation \begin{eqnarray*} \partial_tP + (r-q)S_t \partial_SP + \frac{1}{2}\sigma^2 S_t^2\partial^2_SP dt = rP \end{eqnarray*} This is a diffusion equation (the Black-Scholes equation). In the case where $P$ pays a single cashflow $P(T,S_T)$ at $T$, the Feynman-Kac theorem ensures that the solution to this PDE can be written as an expectation \begin{eqnarray*} P(t,S) &=& \mathbb{E}^\mathbb{Q}_t\left[ e^{-\int_t^T} P(T,Y_T) |Y_t = S\right] \end{eqnarray*} where $(X,\mathbb{Q})$ is any probability space, $W^\mathbb{Q}$ a Brownian motion on it and $Y$ a process satisfying the SDE $$ \frac{dY_t}{Y_t} = (r-q) dt + \sigma dW^\mathbb{Q}_t $$ Such a probability $\mathbb{Q}$ is usuallly called the risk-neutral measure and the process $Y$ is usually written $S$. But they are only mathematical constructs that can make computation easier because the real-world drift is irrelevant. The core of the argument is accounting for the PnL of our strategy and absence of arbitrage. The fact that the price does not depend on the drift is due to the fact that it cancels out when we hold the underlying as our hedge.

  • $\begingroup$ In this writeup, the knowledge about the dividend $q$ is represented in a two rule discrete time stepping representation of the state of $\Pi_t$, and then the stock process is represented without knowledge of $q$ as $dS=S(\mu dt+\sigma dW)$. This form of $dS$ doesn't represent the market's continuous deflation of the price of a dividend-paying stock by the rate of payout. If you only watched the price of $S_t$, you wouldn't see dividend deflation. Other writeups sometimes put $q$ in $dS$ as $dS=S((\mu-q)dt+\sigma dW)$, which will show the deflation but maybe confuse the rule for $d\Pi$. $\endgroup$ – Lars Ericson Aug 18 '17 at 0:41

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